Regular genus and gem-complexity of some mapping tori

Abstract

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms f : M → M for a certain class of PL-manifolds M . These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomor-In particular, for M = S d−1 × S 1 or S d−1 × − S 1 , our construction gives a crystallization of a mapping torus of a (PL) homeomorphism f : M → M with regular genus d 2 − d. As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism f : (S 2 × S 1 ) → (S 2 × S 1 ) with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product RP 3 × S 1 among closed connected prime orientable PL 4-manifolds.MSC 2010 : Primary 57Q15. Secondary 05C15; 57N10; 57N13; 57N15; 57Q05; 55R10.

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

“…In particular, for 3 ≤ d ≤ 5, a lot of classifying results in PL-category have been obtained for both closed connected PL d-manifolds and compact connected PL d-manifolds with boundary (cf. [2,3,10,11,15,16]). In [5], we gave a lower bound for the regular genus of a closed connected PL 4-manifold.…”